#8731111

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#35714	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4792 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2,4,7)(3,6,8,5)(9,10)(11,15)(12,16)(13,14)(17,18)(19,26)(20,23)(21,24)(22,25)(27,29)(28,31)(30,32);(1,3)(2,5)(4,8)(6,7)(10,14)(11,16)(17,19,22,24)(18,21,25,26)(20,27,28,32)(23,30,31,29);(1,3)(2,5)(4,8)(6,7)(9,11)(10,12)(13,16)(14,15)(20,28)(23,31)(27,32)(29,30);(3,8)(5,6)(9,11)(10,12)(13,16)(14,15)(17,20,22,28)(18,23,25,31)(19,27,24,32)(21,30,26,29);(1,2)(3,5)(4,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,18)(19,21)(20,23)(22,25)(24,26)(27,30)(28,31)(29,32);(1,3)(2,6)(4,8)(5,7)(17,21)(18,24)(19,25)(20,29)(22,26)(23,27)(28,30)(31,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8638307);